EP0555493B1 - Electromagnetic flow meter for water in not-full state - Google Patents

Abstract

An electromagnetic flowmeter for water conveyance in a semifull state, whose measurement error caused by the inclination of a pipeline can be reduced. In the flowmeter, excitation coils U and L provided on the upper and lower sides of a pipeline 1 are excited alternately and the flow rate Q0 of the pipeline is given by the equation Q0 = εU0/k0 where εU0 is determined by the ratio of the output voltages of the coils U, L when the coils are excited P0 = εL0/εU0 and k0 is the sensitivity at that time. The flowmeter is characterized in that the data of the relation of the flow rate Q with the output ratio εL/εU and the data of the relation of the flow rate Q with the sensitivity k are measured beforehand using various inclinations α, β, η, δ,... of the pipeline as parameters, and from these data, the flow rate Q0 is calculated and corrected by using the curve representing data on an inclination approximate to an unknown inclination ς of the pipeline.

Description

• The present invention relates to an electromagnetic flow meter for measuring the rate of flow of fluid in a conduit that is not full.
• JP-A-59-230115 describes such an apparatus in which coils are arranged above and below a conduit and are excited alternately to judge on the basis of the respective outputs whether the conduit is full.
• A configuration in which coils arranged in upper and lower portions of a conduit are connected in series is seen in JP-A 52-48356.
• The applicant of the present invention has proposed an electromagnetic flow meter for measuring the rate of flow which is flowing in a conduit in a not-full state on the basis of the theory of electromagnetic flow meter in Japanese Patent Application No. Hei 3-5631. The above proposal was laid open on 31 st August 1993, after the publication of the present application, in JP-A 5-223605).
• The proposed technique has such a configuration as shown in Fig. 1.
• The reference numeral 1 designates a conduit which is circular in section; 2, 2 designate a pair of electrodes which are disposed at laterally symmetrical positions with respect to a vertical line passing the center of the conduit 1; and U and L designate flux coils which are respectively provided at the upper and lower sides of the conduit 1 so as to be excited alternately to produce spatially different and uneven flux distributions in different periods. The reference numeral 4 designates a flow detector having such a structure as mentioned above.
• The reference numeral 5 designates an excitation circuit for alternately exciting the upper and lower excitation coils U and L on the basis of a signal of a timing circuit 6. The reference numeral 7 designates an amplifier for amplifying a voltage induced between the pair of electrodes 2, 2 to form an output voltage, and S1 designates a changeover switch which operates on the basis of the signal of the timing circuit 6. The changeover switch S1 operates in synchronism with a changeover switch S2 which serves to change the excitation period of the two excitation coils U and L, so that the changeover switch S1 is turned to the a side when the upper excitation coil U is excited and the changeover switch S1 is turned to the b side when the lower excitation coil L is excited.
• The reference numerals 8A and 8B designate sample & hold circuits which are supplied with output voltages ε_U and ε_L from the a and b contacts of the changeover switch S1 so as to sample-hold the voltages ε_U and ε_L respectively; 9 designates an A/D conversion circuit for converting analog signals given from the sample & hold circuits 8A and 8B into digital signals respectively; 10 designates a corrective arithmetic circuit having a program for performing a corrective arithmetic operation; and 11 designates an output terminal for outputting a flow rate signal as a result of the arithmetic operation.
• The ratio ε_L/ε_U between the output voltages ε_U and ε_L from the amplifier 1 has a fixed relationship to the water level h. When the water level h and the ratio ε_L/ε_U are respectively taken as the abscissa and ordinate as shown in Fig. 2, one curve A showing the relation between the two is obtained.
• The ratio ε_U/Q between the output voltage ε_U and the real flow rate Q indicates the sensitivity of the flow meter. When the sensitivity is represented by k, the water level h and the sensitivity k have a functional relation expressed by a curve B shown in Fig. 3. The relation between the water level h and the sensitivity k is independent of the gradient of the conduit.
• In Figs. 2 and 3, the water level h as the abscissa is expressed by the ratio of the real water level to the diameter (inner diameter) D of the conduit 1. The curves A and B are preliminarily calculated by measuring the output ratio ε_L/ε_U and the sensitivity k while changing the water level h in a range of from 0 to 1.0D in the condition where the conduit 1 is merely attached to a pipe line while being fixed at a suitable gradient tanΘ as shown in Fig. 4.
• Then, the flow detector 4 is connected to a pipe line where the flow rate is to be measured. If, in this condition, the output ratio ε_L0/ε_U0 thus measured is equal to P₀, then the water level h₀ can be found from the curve A in Fig. 2. Further, the sensitivity k₀ under the condition that the water level is h₀ can be found from the curve B in Fig. 3, so that the true flow rate Q₀ can be calculated as: $${\text{Q}}_{\text{0}}{\text{= ε}}_{\text{U0}}{\text{/k}}_{\text{0}}$$

  

  by the corrective arithmetic circuit 10. This is the gist of the above-mentioned previously-filed patent application.
• In the background technique, the curve A in Fig. 2 has been considered to be independent of the pipe line gradient tanΘ.
• The inventors of the present invention have made eager researches to improve the background technique to thereby improve measurement accuracy. As a result, however, the following fact has been found.
• That is, when the pipe line gradient changes actually, the flow rate distribution changes delicately because the average flow rate changes though the water level may be constant. Accordingly, the curves in Figs. 2 and 3 are respectively shifted according to the change of the pipe line gradient as shown in Figs. 5 and 6. This shifting is the cause of measurement error.
• The water level h has no role but a role as an inclusion term for calculating the sensitivity k on the basis of the output ratio ε_L/ε_U. Further, at any given pipe line gradient, the water level h and the flow rate Q have one-to-one correspondence with each other. Accordingly, in each of Figs. 5 and 6, the abscissa is replaced by the flow rate Q at the gradient used in the measurement.
• That is, when the abscissa is plotted by the flow rate Q as shown in Figs. 5 and 6, not only the curve indicating the relation between the flow rate Q and the output ratio ε_L/ε_U is shifted laterally correspondingly to the gradients α, β and γ as shown in Fig. 5 but the curve indicating the relation between the flow rate Q and the sensitivity k is shifted laterally as shown in Fig. 6.
Relation between pipe line gradient and flow rate.
• It is now assumed that the pipe line gradient ρ is unknown. Further assuming that the output ratio ε_L/ε_U at this time is P₀, first the flow quantities Qα₀, Qβ₀ and Qγ₀ are determined by using a group of curves having α, β and γ as parameters in Fig. 5. Each of the flow quantities Qα₀, Qβ₀ and Qγ₀ indicates that "if the gradient is α (β or γ), the current flow rate Q₀ shows the same water level as that of the flow rate Qα₀ (Qβ₀ or Qγ₀)".
• Accordingly, it is a matter of course that the following relations are established. $${\text{If ρ=α, then Q}}_{\text{0}}{\text{=Qα}}_{\text{0}}\text{.}$$

  

  $${\text{If ρ=β, then Q}}_{\text{0}}{\text{=Qβ}}_{\text{0}}\text{.}$$

  

  $${\text{If ρ=γ, then Q}}_{\text{0}}{\text{=Qγ}}_{\text{0}}\text{.}$$

  
• Because the gradient ρ is however unknown now, the real flow rate Q₀ is obtained by performing the arithmetic calculation of ε_U0/k₀ through the corrective arithmetic circuit 10 after calculating the sensitivity k ₀ from the curves in Fig. 6 with Qα₀, Qβ₀ and Qγ₀ as inclusion terms.
• The thus obtained relations between Q₀ and Qα₀, Qβ₀ and Qγ₀ are as follows. $${\text{If ρ<α, then Q}}_{\text{0}}{\text{<Qα}}_{\text{0}}\text{;}$$

  

  $${\text{If ρ=α, then Q}}_{\text{0}}{\text{=Qα}}_{\text{0}}\text{;}$$

  

  $${\text{If ρ>α, then Q}}_{\text{0}}{\text{>Qα}}_{\text{0}}\text{.}$$

  

  Similarly, the following relations are established. $${\text{If ρ<β, then Q}}_{\text{0}}{\text{<Qβ}}_{\text{0}}\text{;}$$

  

  $${\text{If ρ=β, then Q}}_{\text{0}}\text{=Q;}$$

  

  $${\text{If ρ>β, then Q}}_{\text{0}}{\text{>Qβ}}_{\text{0}}\text{; and so on.}$$

  
• Accordingly, if data corresponding to Figs. 5 and 6 are taken at various gradients α, β, γ, δ ... sufficiently finely spaced and over a sufficient range so that Qα₀, Qβ₀, Qγ₀, Qδ₀, ... can be compared with Q₀ as shown in Figs. 7A, 7B, and 7C, the currently unknown gradient ρ can be known to a corresponding accuracy.
• An object of the present invention is to provide a highly accurate electromagnetic flow meter for a fluid not completely filling a conduit.
• The invention provides an electromagnetic flowmeter for measuring flow rate of a fluid not completely filling a conduit, comprising: coils mounted above and below the measuring conduit for generating a magnetic field across the conduit; means for selectively energising the upper coil or the lower coil; sensor means comprising a pair of electrodes arranged to sense the induced voltage across the conduit caused by the passage of the fluid through the magnetic field; and processing means comprising means arranged:
• (a) to store a first relationship determined in advance between a ratio P of the outputs of the sensor means when the upper and lower coils are respectively energised and the flow rate Q of the fluid, for each of a plurality of known gradients;
• (b) to store a second relationship determined in advance between the sensitivity k, defined as a ratio of the output of the sensor means when the upper coil is energised to the flow rate Q in the conduit, and the flow rate Q, for each of the plurality of known gradients;
• (c) to receive signals ε_U0, ε_L0 from the sensor means;
• (d) to calculate from the received signals ε_U0 and ε_L0 and the stored first and second relationships the actual gradient of the conduit; and
• (e) to calculate a flow rate by using the calculated gradient in the stored relationships.
• The invention also provides a method of measuring flow rate of a fluid not completely filling a conduit, using an electromagnetic flowmeter comprising: coils mounted above and below the measuring conduit for generating a magnetic field across the conduit; means for selectively energising the upper coil or the lower coil; sensor means comprising a pair of electrodes arranged to sense the induced voltage across the conduit caused by the passage of the fluid through the magnetic field; the method comprising the steps of:
• (a) storing a first relationship determined in advance between a ratio P of the outputs of the sensor means when the upper and lower coils are respectively energised and the flow rate Q of the fluid, for each of a plurality of known gradients;
• (b) storing a second relationship determined in advance between the sensitivity k, defined as a ratio of the output of the sensor means when the upper coil is energised to the flow rate Q in the conduit, and the flow rate Q, for each of the plurality of known gradients;
• (c) receiving signals ε_U0, ε_L0 from the sensor means;
• (d) calculating from the received signals ε_U0 and ε_L0 and the stored first and second relationships the actual gradient of the conduit; and
• (e) calculating a flow rate by using the calculated gradient in the stored relationships.
• Because the corrective arithmetic operation can be performed by using curves corresponding to Fig. 5 and 6 measured at gradients most closely approximating to the current gradient ρ (unknown), the current gradient ρ can be found by the method described above.
• For example, Qα₀ and Qβ₀ are first obtained by using curves corresponding to Figs. 5 and 6 measured at gradient α.
• If ρ=α, Q₀ is not equal to the true real flow rate. The relation in size between ρ and α and the relation in size between Q₀ and Qα₀ are however equal to each other as follows: $${\text{If ρ<α, then Q}}_{\text{0}}{\text{<Qα}}_{\text{0}}\text{;}$$

  

  $${\text{If ρ=α, then Q}}_{\text{0}}{\text{=Qα}}_{\text{0}}\text{; and}$$

  

  $${\text{If ρ>α, then Q}}_{\text{0}}{\text{>Qα}}_{\text{0}}\text{.}$$

  
• Accordingly, the current gradient ρ can be found by the method described above.
• More in detail, relations with respect to suitable gradients α and β are selected from the relations (curves) of Figs. 5 and 6 preliminarily measured at various gradients. This selection is performed arbitrarily by an operator. As another method, α and β may be fixed. Alternatively, α and β which have been measured preliminarily may be used directly.
• Provisional real flow quantities Q₀' and Q₀" are calculated on the basis of the equation (1) described above. using the thus selected two curves. Q₀' and Q₀" are preliminarily measured at gradients α and β. The provisional real flow quantities Q₀' and Q₀" have been considered as real flow quantities in the above-mentioned Japanese Patent Application No. Hei-3-5631. It is a matter of course that the provisional real flow quantities are very approximate compared to the true value (real flow rate).
• In this specification, the provisional real flow quantities Q₀' and Q₀" are calculated. The true gradient ρ is calculated by using the provisional real flow quantities Q₀' and Q₀" and the flow quantities Qα₀ and Qβ₀ obtained from Fig. 5.
• When the relations between the true gradient ρ and the selected gradients α and β are ρ<α<β, the relation shown in Fig. 8A is obtained.
• Similarly, in the case of ρ<α<β, the relation shown in Fig. 8B is obtained, and, in the case of α<β<ρ, the relation shown in Fig. 8C is obtained.
• The relations shown in the respective drawings are expressed by the equation: $$\text{ρ =}\frac{{\text{Qβ}}_{\text{0}}{\text{-Q}}_{\text{0}}\text{"}}{{\text{Qβ}}_{\text{0}}{\text{-Qα}}_{\text{0}}{\text{+Q}}_{\text{0}}{\text{'-Q}}_{\text{0}}\text{"}}\text{α +}\frac{{\text{Q}}_{\text{0}}{\text{'-Qα}}_{\text{0}}}{{\text{Qβ}}_{\text{0}}{\text{-Qα}}_{\text{0}}{\text{+Q}}_{\text{0}}{\text{'-Q}}_{\text{0}}\text{"}}\text{β}$$

  
• When the gradient ρ becomes known as described above, an exact errorless flow rate is calculated by performing a corrective arithmetic operation by using curves measured at a gradient most nearly approximate to the gradient ρ or by using suitable curves obtained by interpolation/extrapolation from the measured curves.
BRIEF DESCRIPTION OF DRAWINGS
• Fig. 1 is a block diagram of a previously proposed electromagnetic flow meter for water in a not-full state;
• Fig. 2 is a graph of the water level versus the output ratio;
• Fig. 3 is a graph of the water level versus the sensitivity;
• Fig. 4 is a schematic view for explaining the pipe line gradient;
• Fig. 5 is a graph of the flow rate versus the output;
• Fig. 6 is a graph of the flow rate versus the sensitivity;
• Figs. 7A through 7C are graphs of the flow rate versus the sensitivity ratio showing different relations in size between the measured gradient ρ and a gradient α for which data have been preliminarily stored;
• Fig. 8A is a graph showing the relation between the gradient in the case of α<ρ<β and the flow rate difference ΔQ;
• Fig. 8B is a graph showing the relation between the gradient in the case of ρ<α<β and the flow rate difference ΔQ;
• Fig. 8C is a graph showing the relation between the gradient in the case of α<β<ρ and the flow rate difference ΔQ;
• Fig. 9A is a front view of a flow detector, Fig. 9B is a section taken along the line A-A of Fig. 9A, and Fig. 9C is a block diagram of a flow meter as an embodiment of the present invention;
• Fig. 10 is a schematic view of a water passing equipment;
• Figs. 11 and 12 are graphs of instrumental error;
• Fig. 13 is a flow chart showing a process in the embodiment of the invention; and
• Fig. 14 is a perspective view of a coil used in the embodiment.
• A flow detector 4 as shown in Figs. 9A and 9B is produced by way of trial, and an electromagnetic flow meter for water in a not-full state as shown in the block diagram of Fig. 9C is formed. Data of curves corresponding to Figs. 5 and 6 are sampled in advance for a pipe line 13 with its gradient set to 2/1000 and 6/1000 by using a water passing equipment shown in Fig. 10, and the sampled data are stored in a memory 100.
• In Fig. 9C, members the same as those in Fig. 1 are correspondingly referenced and the description is partly omitted.
• Fig. 11 is a graph of instrumental error in the case where a corrective arithmetic operation is performed by providing, for use as reference curves, curves measured at current gradients 2/1000 and 6/1000 and selecting, by the method of the invention, the curves measured at the gradient that is judged to be the closer approximation to the unknown gradient of the conduit. The real gradient of the conduit which was the subject of the measurement was 0/1000. The instrumental error is expressed in % as $$\text{100 x (measured value - true value)/true value.}$$

  
• The true value is obtained by a reference flow meter.
• Fig. 12 shows instrumental error in the case where the flow rate is measured by reference to curves corresponding to Figs. 5 and 6 preliminarily measured at the gradient 6/1000. The instrumental error in Fig. 12 is relatively large because the difference between the gradient 6/1000 used at the time of sampling the reference curve data and the gradient 0/1000 at the time of the measurement is large.
• The effect of the invention will becomes clearer from comparison between Figs. 11 and 12. The measurement accuracy can be improved more greatly by increasing the number and range of the reference curves.
• The operation of the apparatus in this embodiment is shown in the flow chart of Fig. 13.
• In a step S1, P₀=ε_L0/ε_U0 is obtained by the output voltages ε_U0 and ε_L0 of the amplifier 1, that is, by the ratio of the lower coil output to the upper coil output. The calculation of the ratio is performed by the corrective arithmetic circuit 10. A CPU for exclusive use is used as the circuit 10.
• Then, in a stepS3, the circuit 10 reads the relations of Fig. 5 at the gradients 2/1000 and 6/1000 from the memory 100, obtains flow rates Q_2/1000 and Q_6/1000 at the respective gradients correspondingly to P₀ and stores the flow rates in registers.
• In a step S5, the circuit 10 reads the relations of Fig. 6 at the respective gradients from the memory 100, obtains provisional real flow rates Q'(2/1000) and Q"(6/1000) by using the equation 1 and stores the provisional real flow rates in registers.
• In a step S7, the equation (2) is executed by using the values obtained in the steps S3 and S5. As a result, the true gradient ρ is found as about 0/1000.
• Accordingly, in a step 9, the gradient 2/1000 more nearly approximating to the true gradient ρ is selected. In a step S11, the real flow rate Q₀ is calculated by using the relations of Figs. 5 and 6 at the gradient 2/1000. In this embodiment, the real flow rate is as follows. $${\text{Q}}_{\text{0}}\text{= Q'(2/1000)}$$

  
• Although the above-mentioned embodiment has shown the case where two gradients are preliminarily obtained and used as the relations of Figs. 5 and 6, a larger number of data may be preliminarily obtained to improve the measurement accuracy further. For example, data may be obtained preliminarily at gradient intervals of 1/1000.
• It is preferable that a gradient difference of about 1/1000 to about 10/1000 is given to the gradients α and β used for calculating the true gradient.
• The inner diameter of the pipe line in this embodiment is 240 mm.
• The electrodes 2 have a width of 40 mm in the direction of flow, an opening angle of 90° and a thickness of 2 mm.
• The form and size of the upper and lower coils are shown in Fig. 14.
• It should be added that the coils as used are each composed of a winding of 1300 turns.
• Because the electromagnetic flow meter for water in a not-full state according to the present invention is formed as described above, an accurate flow rate can be obtained by a corrective arithmetic operation to correct the influence of the pipe line gradient. Accordingly, the flow meter contributes to an improvement in accuracy of the electromagnetic flow meter for water in a not-full state.
• Because the measurement error caused by the gradient difference is reduced, the range of gradients at which the pipe line may be mounted can be widened while keeping the accuracy within a certain range.

Claims (8)

1. An electromagnetic flowmeter for measuring flow rate of a fluid not completely filling a conduit, comprising: coils (U,L) mounted above and below the measuring conduit (1) for generating a magnetic field across the conduit; means (5,S2) for selectively energising the upper coil (U) or the lower coil (L); sensor means comprising a pair of electrodes (2) arranged to sense the induced voltage across the conduit caused by the passage of the fluid through the magnetic field; and processing means (9,10,100) comprising means arranged:

   (a) to store (100) a first relationship determined in advance between a ratio P of the outputs of the sensor means when the upper and lower coils are respectively energised and the flow rate Q of the fluid, for each of a plurality of known gradients (α,β...);

   (b) to store a second relationship determined in advance between the sensitivity k, defined as a ratio of the output of the sensor means when the upper coil is energised to the flow rate Q in the conduit, and the flow (α,β...); rate Q, for each of the plurality of known gradients

   (c) to receive signals ε_U0, ε_L0 from the sensor means (5,S2);

   (d) to calculate (S1 to S9) from the received signals ε_U0 and ε_L0 and the stored first and second relationships the actual gradient of the conduit; and

   (e) to calculate a flow rate by using the calculated gradient in the stored relationships.
2. An electromagnetic flowmeter according to claim 1, comprising means arranged:

   (f) to calculate (S1) a measurement ratio (P₀) between the signals ε_U0 and ε_L0;

   (g) to read (S3) said calculated measurement ratio (P₀) into the stored first relationship to calculate flow rates Q _2/1000 , Q _6/1000 for the fluid in the measuring conduit for two of the said known gradients;

   (h) to read (S5) said calculated flow rates Q _2/1000 , Q _6/1000 into the stored second relationship to calculate a provisional real flow rate (Q', Q") in said measuring conduit for each of said two gradients;

   (i) to determine (S7) a real gradient (ρ) for the measuring conduit in accordance with the calculated flow rates and provisional real flow rates determined in steps

   (g) and (h);

   (j) to select (S9) a gradient that approximates to the determined real gradient (ρ); and

   (k) to repeat (S11) steps (g) and (h) using the said selected gradient to calculate a real flow rate.
3. An electromagnetic flowmeter according to claim 2, comprising means arranged to determine the real gradient (ρ) in step (i) in accordance with the equation: $$\text{ρ =}\frac{{\text{Qβ}}_{\text{0}}{\text{-Q}}_{\text{0}}\text{"}}{{\text{Qβ}}_{\text{0}}{\text{-Qα}}_{\text{0}}{\text{+Q}}_{\text{0}}{\text{'-Q}}_{\text{0}}\text{"}}\text{α +}\frac{{\text{Q}}_{\text{0}}{\text{'-Qα}}_{\text{0}}}{{\text{Qβ}}_{\text{0}}{\text{-Qα}}_{\text{0}}{\text{+Q}}_{\text{0}}{\text{'-Q}}_{\text{0}}\text{"}}\text{β}$$

   
4. An electromagnetic flowmeter according to claim 2 or claim 3, comprising means arranged to select as the gradient in step (j) the known gradient (α,β) closest to the real gradient (ρ).
5. A method of measuring flow rate of a fluid not completely filling a conduit, using an electromagnetic flowmeter comprising: coils (U,L) mounted above and below the measuring conduit (1) for generating a magnetic field across the conduit; means (5,S2) for selectively energising the upper coil (U) or the lower coil (L); sensor means comprising a pair of electrodes (2) arranged to sense the induced voltage across the conduit caused by the passage of the fluid through the magnetic field; the method comprising the steps of:

   (a) storing (100) a first relationship determined in advance between a ratio P of the outputs of the sensor means when the upper and lower coils are respectively energised and the flow rate Q of the fluid, for each of a plurality of known gradients (α,β...);

   (b) storing a second relationship determined in advance between the sensitivity k, defined as a ratio of the output of the sensor means when the upper coil is energised to the flow rate Q in the conduit, and the flow rate Q, for each of the plurality of known gradients (α,β...);

   (c) receiving signals ε_U0, ε_L0 from the sensor means (5,S2);

   (d) calculating (S1 to S9) from the received signals ε_U0 and ε_L0 and the stored first and second relationships the actual gradient of the conduit; and

   (e) calculating a flow rate by using the calculated gradient in the stored relationships.
6. A method according to claim 5, comprising the steps of:

   (f) calculating (S1) a measurement ratio (P₀) between the signals ε_U0 and ε_L0;

   (g) reading (S3) said calculated measurement ratio (P₀) into the stored first relationship to calculate flow rates Q _2/1000 , Q _6/1000 for the fluid in the measuring conduit for two of the said known gradients;

   (h) reading (S5) said calculated flow rates Q _2/1000 , Q _6/1000 into the stored second relationship to calculate a provisional real flow rate (Q', Q") in said measuring conduit for each of said two gradients;

   (i) determining (S7) a real gradient (ρ) for the measuring conduit in accordance with the calculated flow rates and provisional real flow rates determined in steps (g) and (h);

   (j) selecting (S9) a gradient that approximates to the determined real gradient (ρ); and

   (k) repeating (S11) steps (g) and (h) using the said selected gradient to calculate a real flow rate.
7. A method according to claim 6, wherein in step (i) the real gradient (ρ) is determined in accordance with the equation: $$\text{ρ =}\frac{{\text{Qβ}}_{\text{0}}{\text{-Q}}_{\text{0}}\text{"}}{{\text{Qβ}}_{\text{0}}{\text{-Qα}}_{\text{0}}{\text{+Q}}_{\text{0}}{\text{'-Q}}_{\text{0}}\text{"}}\text{α +}\frac{{\text{Q}}_{\text{0}}{\text{'-Qα}}_{\text{0}}}{{\text{Qβ}}_{\text{0}}{\text{-Qα}}_{\text{0}}{\text{+Q}}_{\text{0}}{\text{'-Q}}_{\text{0}}\text{"}}\text{β}$$

   
8. A method according to claim 6 or claim 7, wherein in step (j) the gradient selected is the known gradient (α,β) closest to the real gradient (ρ).

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